Local piecewise polynomial curve of variable order and its appliance for calculating of nodal derivatives

In this paper, we present a new Hermite type curve piecewise polynomial of C1 continuity on nonuniform grids. Depending on the location of the knots of the grid (uniform and nonuniform), the curve is quadratic or cubic. The approximation error is investigated. For nonuniform grids a method for calculating the first and second derivatives in knots is proposed. We also perform several numerical experiments which exemplify the properties of the proposed curve. Graphic examples are given.